Optimal control for absolutely continuous stochastic processes and the mass transportation problem
This paper deals with the optimal control problem for \(\mathbb{R}^d\)-valued absolutely continuous stochastic processes with given marginal distributions that arises in the mass transportation problem. Using the weak convergence method and the construction of a Markov diffusion process from a marginal distribution, the author proves the following results under appropriate conditions. 1) For \(d=1\), there exists a unique minimizer which is a non-random function of a time and an initial point. 2. For \(d> 1\), there exists a minimizer and for any minimizer \(X(\cdot)\), \(b^X(t, x):= E[(dX(t)/dt)/(t, X(t)= x)]\) does not depend on \(X(\cdot)\). Namely, minimizers satisfy the same ordinary differential equation. Moreover, applying the results to \(d= 1\), he gives a positive answer to Salisbury's problem.
absolutely continuous stochastic process
scientific article; zbMATH DE number 1873771